A submartingale need not have increasing sample paths

The measure $P=\frac{1}{2}{\delta }_{-1}+\frac{1}{2}{\delta }_1$ is a measure of total mass one. The displayed filtration is increasing. All $X_n$ are adapted and bounded by one, hence integrable. On the only two events of ${\mathcal{F}}_0$, the zero function has the same integrals as $X_1$, since $E{X}_1=(-1+1)\mathrm{/}2=0$. It is therefore its conditional version. At later times $X_{n+1}=X_n$ is known, so $E[X_{n+1}\mid {\mathcal{F}}_n]=X_n$. Thus $X$ is a martingale and hence a submartingale Martingale submartingale and supermartingale.

On the measurable atom $\{-1\}$, whose probability is $1\mathrm{/}2$, one has $X_1=-1<0=X_0$. Any exceptional set outside which paths are nondecreasing must contain this atom and therefore cannot have probability zero. This refutes even almost-sure nondecreasing paths. AC is inherited from the CE class convention; the finite averages supply the displayed versions explicitly.